In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.

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In calculus, the comparison test for series typically consists of a pair of statements about infinite series with non-negative (real-valued) terms:[1]

If the infinite series ∑bn{\displaystyle \sum b_{n}} converges and 0≤an≤bn{\displaystyle 0\leq a_{n}\leq b_{n}} for all sufficiently large n (that is, for all n>N{\displaystyle n>N} for some fixed value N), then the infinite series ∑an{\displaystyle \sum a_{n}} also converges.

If the infinite series ∑bn{\displaystyle \sum b_{n}} diverges and 0≤bn≤an{\displaystyle 0\leq b_{n}\leq a_{n}} for all sufficiently large n, then the infinite series ∑an{\displaystyle \sum a_{n}} also diverges.

Note that the series having larger terms is sometimes said to dominate (or eventually dominate) the series with smaller terms.[2]

If the infinite series ∑bn{\displaystyle \sum b_{n}} is absolutely convergent and |an|≤|bn|{\displaystyle |a_{n}|\leq |b_{n}|} for all sufficiently large n, then the infinite series ∑an{\displaystyle \sum a_{n}} is also absolutely convergent.

If the infinite series ∑bn{\displaystyle \sum b_{n}} is not absolutely convergent and |bn|≤|an|{\displaystyle |b_{n}|\leq |a_{n}|} for all sufficiently large n, then the infinite series ∑an{\displaystyle \sum a_{n}} is also not absolutely convergent.

Note that in this last statement, the series ∑an{\displaystyle \sum a_{n}} could still be conditionally convergent; for real-valued series, this could happen if the an are not all nonnegative.

The second pair of statements are equivalent to the first in the case of real-valued series because ∑cn{\displaystyle \sum c_{n}} converges absolutely if and only if ∑|cn|{\displaystyle \sum |c_{n}|}, a series with nonnegative terms, converges.

Sn{\displaystyle S_{n}} is a nondecreasing sequence and Sn+(T−Tn){\displaystyle S_{n}+(T-T_{n})} is nonincreasing.
Given m,n>N{\displaystyle m,n>N} then both Sn,Sm{\displaystyle S_{n},S_{m}} belong to the interval [SN,SN+(T−TN)]{\displaystyle [S_{N},S_{N}+(T-T_{N})]}, whose length T−TN{\displaystyle T-T_{N}} decreases to zero as N{\displaystyle N} goes to infinity.
This shows that (Sn)n=1,2,…{\displaystyle (S_{n})_{n=1,2,\ldots }} is a Cauchy sequence, and so must converge to a limit. Therefore, ∑an{\displaystyle \sum a_{n}} is absolutely convergent.

The comparison test for integrals may be stated as follows, assuming continuous real-valued functions f and g on [a,b){\displaystyle [a,b)} with b either +∞{\displaystyle +\infty } or a real number at which f and g each have a vertical asymptote:[4]

Another test for convergence of real-valued series, similar to both the direct comparison test above and the ratio test, is called the ratio comparison test:[5]

If the infinite series ∑bn{\displaystyle \sum b_{n}} converges and an>0{\displaystyle a_{n}>0}, bn>0{\displaystyle b_{n}>0}, and an+1an≤bn+1bn{\displaystyle {\frac {a_{n+1}}{a_{n}}}\leq {\frac {b_{n+1}}{b_{n}}}} for all sufficiently large n, then the infinite series ∑an{\displaystyle \sum a_{n}} also converges.

If the infinite series ∑bn{\displaystyle \sum b_{n}} diverges and an>0{\displaystyle a_{n}>0}, bn>0{\displaystyle b_{n}>0}, and an+1an≥bn+1bn{\displaystyle {\frac {a_{n+1}}{a_{n}}}\geq {\frac {b_{n+1}}{b_{n}}}} for all sufficiently large n, then the infinite series ∑an{\displaystyle \sum a_{n}} also diverges.